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Corollary to Hahn-Banach Thm

  1. May 23, 2013 #1
    1. The problem statement, all variables and given/known data
    To clarify- this isn't a homework problem; its something that's stated as a corollary in my notes (as in the proof is supposed to be obvious) and I haven't yet managed to prove it- I'm probably just missing something! Would appreciate a hint or a link to where I might find the proof.

    Let X be a normed vector space, t [itex]\in[/itex] X such that for all g [itex]\in[/itex] X', g(t) = 0. Then t = 0.

    2. Relevant equations
    Hahn-Banach Theorem (stated on my course as:)
    Let M be a subspace of a normed vector space X. Let f [itex]\in[/itex] M' . Then there exists g [itex]\in[/itex] X' such that the norm of f (wrt M') is equal to the norm of g (wrt X'), and g is equal to f on M.


    3. The attempt at a solution
    Just a bit confused- this is equivalent to, the only point all linear functionals can vanish at is zero. Presumably I want to define some linear subspace to then use but the only one that jumps out at me is span(t) which then doesn't seem to give me anything. Any hints?
     
  2. jcsd
  3. May 23, 2013 #2

    micromass

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    You're going in the right direction. Assume that ##t\neq 0##. Then set ##M= span(t)##. Now define a suitable nonzero functional on ##M## and extend it by Hahn-Banach.
     
  4. May 23, 2013 #3
    Possibly in the right direction, but unfortunately not at speed :P

    I want to find a linear functional f defined on M s.t. f vanishes at t (so that its Hahn-Banach extension will satisfy the given property).. but if f vanishes at t then it vanishes on the whole of M, which is not particularly useful. I'm probably being incredibly slow but would appreciate a further hint, sorry! Also I have a lot of exams to revise for and don't really want to spend loads of time trying to prove this one little bit.... Thank you in advance for any help!!
     
  5. May 23, 2013 #4

    micromass

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    You want to assume that ##t\neq 0## and you want to find a linear functional that does not vanish at ##t##. This would be a contradiction with the property that ##t## has (namely, that all the linear functionals vanish there.

    So, can you find a linear functional ##f:M\rightarrow \mathbb{R}## with ##M=span(t)## such that ##f(t)\neq 0##?
     
  6. May 24, 2013 #5
    Hmmn I think I was getting confused with the logic of what I was trying to do then. I can take f(at) = a |t| (where |t| is its norm). Then if every functional vanishes at t then the extension of f, g satisfies g(t) = 0 = |t| and by positive definiteness t is zero.
     
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